[Paper Review] Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes

Alexander I. Bobenko, Ivan Izmestiev|arXiv (Cornell University)|Sep 15, 2006

Computational Geometry and Mesh Generation13 references4 citations

TL;DR

This paper presents a constructive proof of Alexandrov's theorem using the total scalar curvature functional (Regge action) on generalized convex polytopes, showing its non-degeneracy via a generalized Alexandrov-Fenchel inequality. The key result is a novel algorithmic construction of a convex polytope from a given boundary metric, implemented computationally, with uniqueness and existence established through Hessian analysis and weighted Delaunay triangulations of polyhedral surfaces.

ABSTRACT

We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by generalizing the Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.

Motivation & Objective

• To provide a constructive proof of Alexandrov’s theorem on the existence and uniqueness of a convex polytope with a given boundary metric.
• To establish a connection between generalized convex polytopes and weighted Delaunay triangulations of polyhedral surfaces.
• To prove the non-degeneracy of the Hessian of the total scalar curvature functional on positively curved generalized convex polytopes.
• To generalize the Alexandrov-Fenchel inequality to support the non-degeneracy result.
• To implement the construction algorithm in a computer program for practical realization.

Proposed method

• Model generalized convex polytopes as simplicial complexes formed by gluing pyramids over triangles of a geodesic triangulation of the boundary surface.
• Represent each generalized polytope by a pair (T, r), where T is the triangulation and r_i are the lengths of the side edges of the pyramids.
• Define the total scalar curvature functional as the Regge action: ∑ℓ_e κ_e, where ℓ_e is edge length and κ_e is the angle deficit at edge e.
• Show that the Hessian of the total scalar curvature functional equals the Hessian of the volume of the dual generalized polyhedron.
• Prove non-degeneracy of the Hessian by generalizing the Alexandrov-Fenchel inequality to the setting of polyhedral surfaces.
• Use the non-degeneracy to construct a deformation path from a given metric to a convex polytope via gradient flow of the scalar curvature functional.

Experimental results

Research questions

• RQ1 Can Alexandrov’s existence and uniqueness theorem for convex polytopes be proven constructively using geometric functionals?
• RQ2 What is the relationship between generalized convex polytopes and weighted Delaunay triangulations of polyhedral surfaces?
• RQ3 How does the Hessian of the total scalar curvature functional relate to the geometry of the dual polyhedron?
• RQ4 Can the Alexandrov-Fenchel inequality be extended to the discrete setting of polyhedral surfaces to ensure non-degeneracy?
• RQ5 Is there an algorithmic method to reconstruct a convex polytope from its intrinsic boundary metric?

Key findings

• The total scalar curvature functional (Regge action) is strictly convex on the space of generalized convex polytopes, ensuring a unique minimizer.
• The Hessian of the total scalar curvature functional is non-degenerate, which implies the existence of a unique deformation path to a convex polytope.
• The Hessian of the total scalar curvature is equivalent to the Hessian of the volume of the dual generalized polyhedron, establishing a duality between curvature and volume functionals.
• A generalized version of the Alexandrov-Fenchel inequality holds for polyhedral surfaces, proving the non-degeneracy of the Hessian.
• The deformation path from a given metric to a convex polytope is constructed via gradient flow of the scalar curvature functional, with convergence guaranteed by strict convexity.
• The algorithmic construction is implemented in a computer program, enabling explicit computation of the polytope from a given boundary metric.